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Old page wikitext, before the edit (old_wikitext) | 'In mathematics, specifically in [[complex analysis]], '''Cauchy's estimate''' gives local bounds for the [[derivative]]s of a [[holomorphic function]]. These bounds are optimal.
== Statement and consequence==
Let <math>f</math> be a holomorphic function on the open ball <math>B(a, r)</math> in <math>\mathbb C</math>. If <math>M</math> is the sup of <math>|f|</math> over <math>B(a, r)</math>, then Cauchy's estimate says:<ref>{{harvnb|Rudin|1986|loc=Theorem 10.26.}}</ref> for each integer <math>n > 0</math>,
:<math>|f^{(n)}(a)| \le \frac{n!}{r^n} M</math>
where <math>f^{(n)}</math> is the ''n''-th [[complex derivative]] of <math>f</math>; i.e., <math>f' = \frac{\partial f}{\partial z}</math> and <math>f^{(n)} = (f^{(n-1)})^'</math> (see {{section link| Wirtinger_derivatives|Relation_with_complex_differentiation}}).
Moreover, taking <math>f(z) = z^n, a = 0, r = 1</math> shows the above estimate cannot be improved.
As a corollary, for example, we obtain [[Liouville's theorem (complex analysis)|Liouville's theorem]], which says a bounded entire function is constant (indeed, let <math>r \to \infty</math> in the estimate.) Slightly more generally, if <math>f</math> is an entire function bounded by <math>A + B|z|^k</math> for some constants <math>A, B</math> and some integer <math>k > 0</math>, then <math>f</math> is a polynomial.<ref>{{harvnb|Rudin|1986|loc=Ch 10. Exercise 4.}}</ref>
== Proof ==
We start with [[Cauchy's integral formula]] applied to <math>f</math>
:<math>f(z) = \frac{1}{2\pi i} \int_{|w-a| = r'} \frac{f(w)}{w - z} \, dw,</math>
where <math>r' < r</math>. By the [[differentiation under the integral sign]] (in the complex variable),<ref>This step is Exercise 7 in Ch. 10. of {{harvnb|Rudin|1986}}</ref> we get:
:<math>f^{(n)}(z) = \frac{n!}{2\pi i} \int_{|w-a| = r'} \frac{f(w)}{(w - z)^{n+1}} \, dw.</math>
Thus,
:<math>|f^{(n)}(a)| \le \frac{n!M}{2\pi} \int_{|w-a| = r'} \frac{|dw|}{|w - a|^{n+1}} = \frac{n!M}{{r'}^n}.</math>
Letting <math>r' \to r</math> finishes the proof. <math>\square</math>
== See also ==
*[[Taylor's theorem]]
== References ==
{{reflist}}
*{{Citation |author =[[Lars Hörmander]] |date=1990 |orig-year=1966 |title=An Introduction to Complex Analysis in Several Variables |edition=3rd |publisher=North Holland |isbn=978-1-493-30273-4 |url={{Google books|MaM7AAAAQBAJ|An Introduction to Complex Analysis in Several Variables|plainurl=yes}}}}
* {{cite book | last=Rudin | first=Walter | authorlink = Walter Rudin | title = Real and Complex Analysis (International Series in Pure and Applied Mathematics) | publisher=McGraw-Hill | year=1986 |isbn=978-0-07-054234-1}}
== Further reading ==
* https://math.stackexchange.com/questions/114349/how-is-cauchys-estimate-derived/114363
{{analysis-stub}}' |
New page wikitext, after the edit (new_wikitext) | 'In mathematics, specifically in [[complex analysis]], '''Cauchy's estimate''' gives local bounds for the [[derivative]]s of a [[holomorphic function]]. These bounds are optimal.
== Statement and consequence==
Let <math>f</math> be a holomorphic function on the open ball <math>B(a, r)</math> in <math>\mathbb C</math>. If <math>M</math> is the sup of <math>|f|</math> over <math>B(a, r)</math>, then Cauchy's estimate says:<ref>{{harvnb|Rudin|1986|loc=Theorem 10.26.}}</ref> for each integer <math>n > 0</math>,
:<math>|f^{(n)}(a)| \le \frac{n!}{r^n} M</math>
where <math>f^{(n)}</math> is the ''n''-th [[complex derivative]] of <math>f</math>; i.e., <math>f' = \frac{\partial f}{\partial z}</math> and <math>f^{(n)} = (f^{(n-1)})^'</math> (see {{section link| Wirtinger_derivatives|Relation_with_complex_differentiation}}).
Moreover, taking <math>f(z) = z^n, a = 0, r = 1</math> shows the above estimate cannot be improved.
As a corollary, for example, we obtain [[Liouville's theorem (complex analysis)|Liouville's theorem]], which says a bounded entire function is constant (indeed, let <math>r \to \infty</math> in the estimate.) Slightly more generally, if <math>f</math> is an entire function bounded by <math>A + B|z|^k</math> for some constants <math>A, B</math> and some integer <math>k > 0</math>, then <math>f</math> is a polynomial.<ref>{{harvnb|Rudin|1986|loc=Ch 10. Exercise 4.}}</ref>
== Proof ==
We start with [[Cauchy's integral formula]] applied to <math>f</math>
:<math>f(z) = \frac{1}{2\pi i} \int_{|w-a| = r'} \frac{f(w)}{w - z} \, dw,</math>
where <math>r' < r</math>. By the [[differentiation under the integral sign]] (in the complex variable),<ref>This step is Exercise 7 in Ch. 10. of {{harvnb|Rudin|1986}}</ref> we get:
:<math>f^{(n)}(z) = \frac{n!}{2\pi i} \int_{|w-a| = r'} \frac{f(w)}{(w - z)^{n+1}} \, dw.</math>
Thus,
:<math>|f^{(n)}(a)| \le \frac{n!M}{2\pi} \int_{|w-a| = r'} \frac{|dw|}{|w - a|^{n+1}} = \frac{n!M}{{r'}^n}.</math>
Letting <math>r' \to r</math> finishes the proof. <math>\square</math>
== Related estimate ==
Here is a somehow more general but slightly less precise estimate. It says:<ref>{{harvnb|Hörmander|1990}}</ref> given an open subset <math>U \subset \mathbb{C}</math>, a compact subset <math>K \subset U</math> and an integer <math>n > 0</math>, there is a constant <math>C_{K, n}</math> such that
:<math>\sup_{K} |f^{(n)}| \le C_{K, n} \int_U |f| \, d\mu</math>
where <math>d\mu</math> is the Lebesgue measure.
== See also ==
*[[Taylor's theorem]]
== References ==
{{reflist}}
*{{Citation |author =[[Lars Hörmander]] |date=1990 |orig-year=1966 |title=An Introduction to Complex Analysis in Several Variables |edition=3rd |publisher=North Holland |isbn=978-1-493-30273-4 |url={{Google books|MaM7AAAAQBAJ|An Introduction to Complex Analysis in Several Variables|plainurl=yes}}}}
* {{cite book | last=Rudin | first=Walter | authorlink = Walter Rudin | title = Real and Complex Analysis (International Series in Pure and Applied Mathematics) | publisher=McGraw-Hill | year=1986 |isbn=978-0-07-054234-1}}
== Further reading ==
* https://math.stackexchange.com/questions/114349/how-is-cauchys-estimate-derived/114363
{{analysis-stub}}' |
Unified diff of changes made by edit (edit_diff) | '@@ -18,4 +18,9 @@
:<math>|f^{(n)}(a)| \le \frac{n!M}{2\pi} \int_{|w-a| = r'} \frac{|dw|}{|w - a|^{n+1}} = \frac{n!M}{{r'}^n}.</math>
Letting <math>r' \to r</math> finishes the proof. <math>\square</math>
+
+== Related estimate ==
+Here is a somehow more general but slightly less precise estimate. It says:<ref>{{harvnb|Hörmander|1990}}</ref> given an open subset <math>U \subset \mathbb{C}</math>, a compact subset <math>K \subset U</math> and an integer <math>n > 0</math>, there is a constant <math>C_{K, n}</math> such that
+:<math>\sup_{K} |f^{(n)}| \le C_{K, n} \int_U |f| \, d\mu</math>
+where <math>d\mu</math> is the Lebesgue measure.
== See also ==
' |
New page size (new_size) | 3198 |
Old page size (old_size) | 2762 |
Size change in edit (edit_delta) | 436 |
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0 => '',
1 => '== Related estimate ==',
2 => 'Here is a somehow more general but slightly less precise estimate. It says:<ref>{{harvnb|Hörmander|1990}}</ref> given an open subset <math>U \subset \mathbb{C}</math>, a compact subset <math>K \subset U</math> and an integer <math>n > 0</math>, there is a constant <math>C_{K, n}</math> such that',
3 => ':<math>\sup_{K} |f^{(n)}| \le C_{K, n} \int_U |f| \, d\mu</math>',
4 => 'where <math>d\mu</math> is the Lebesgue measure.'
] |
Lines removed in edit (removed_lines) | [] |
Parsed HTML source of the new revision (new_html) | '<div class="mw-content-ltr mw-parser-output" lang="en" dir="ltr"><p>In mathematics, specifically in <a href="/https/en.wikipedia.org/wiki/Complex_analysis" title="Complex analysis">complex analysis</a>, <b>Cauchy's estimate</b> gives local bounds for the <a href="/https/en.wikipedia.org/wiki/Derivative" title="Derivative">derivatives</a> of a <a href="/https/en.wikipedia.org/wiki/Holomorphic_function" title="Holomorphic function">holomorphic function</a>. These bounds are optimal.
</p>
<div id="toc" class="toc" role="navigation" aria-labelledby="mw-toc-heading"><input type="checkbox" role="button" id="toctogglecheckbox" class="toctogglecheckbox" style="display:none" /><div class="toctitle" lang="en" dir="ltr"><h2 id="mw-toc-heading">Contents</h2><span class="toctogglespan"><label class="toctogglelabel" for="toctogglecheckbox"></label></span></div>
<ul>
<li class="toclevel-1 tocsection-1"><a href="#Statement_and_consequence"><span class="tocnumber">1</span> <span class="toctext">Statement and consequence</span></a></li>
<li class="toclevel-1 tocsection-2"><a href="#Proof"><span class="tocnumber">2</span> <span class="toctext">Proof</span></a></li>
<li class="toclevel-1 tocsection-3"><a href="#Related_estimate"><span class="tocnumber">3</span> <span class="toctext">Related estimate</span></a></li>
<li class="toclevel-1 tocsection-4"><a href="#See_also"><span class="tocnumber">4</span> <span class="toctext">See also</span></a></li>
<li class="toclevel-1 tocsection-5"><a href="#References"><span class="tocnumber">5</span> <span class="toctext">References</span></a></li>
<li class="toclevel-1 tocsection-6"><a href="#Further_reading"><span class="tocnumber">6</span> <span class="toctext">Further reading</span></a></li>
</ul>
</div>
<div class="mw-heading mw-heading2"><h2 id="Statement_and_consequence">Statement and consequence</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/https/en.wikipedia.org/w/index.php?title=Cauchy%27s_estimate&veaction=edit&section=1" title="Edit section: Statement and consequence" class="mw-editsection-visualeditor"><span>edit</span></a><span class="mw-editsection-divider"> | </span><a href="/https/en.wikipedia.org/w/index.php?title=Cauchy%27s_estimate&action=edit&section=1" title="Edit section's source code: Statement and consequence"><span>edit source</span></a><span class="mw-editsection-bracket">]</span></span></div>
<p>Let <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>f</mi>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle f}</annotation>
</semantics>
</math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/132e57acb643253e7810ee9702d9581f159a1c61" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.279ex; height:2.509ex;" alt="{\displaystyle f}"></span> be a holomorphic function on the open ball <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle B(a,r)}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>B</mi>
<mo stretchy="false">(</mo>
<mi>a</mi>
<mo>,</mo>
<mi>r</mi>
<mo stretchy="false">)</mo>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle B(a,r)}</annotation>
</semantics>
</math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2d65752bb841eb20bc3668a1fcdc08a01a3c2c54" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.886ex; height:2.843ex;" alt="{\displaystyle B(a,r)}"></span> in <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {C} }">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mrow class="MJX-TeXAtom-ORD">
<mi mathvariant="double-struck">C</mi>
</mrow>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle \mathbb {C} }</annotation>
</semantics>
</math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f9add4085095b9b6d28d045fd9c92c2c09f549a7" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.678ex; height:2.176ex;" alt="{\displaystyle \mathbb {C} }"></span>. If <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle M}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>M</mi>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle M}</annotation>
</semantics>
</math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f82cade9898ced02fdd08712e5f0c0151758a0dd" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.442ex; height:2.176ex;" alt="{\displaystyle M}"></span> is the sup of <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle |f|}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mrow class="MJX-TeXAtom-ORD">
<mo stretchy="false">|</mo>
</mrow>
<mi>f</mi>
<mrow class="MJX-TeXAtom-ORD">
<mo stretchy="false">|</mo>
</mrow>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle |f|}</annotation>
</semantics>
</math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/940e58aa437f429f47fd743a819e41fedaa1ff9b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:2.572ex; height:2.843ex;" alt="{\displaystyle |f|}"></span> over <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle B(a,r)}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>B</mi>
<mo stretchy="false">(</mo>
<mi>a</mi>
<mo>,</mo>
<mi>r</mi>
<mo stretchy="false">)</mo>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle B(a,r)}</annotation>
</semantics>
</math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2d65752bb841eb20bc3668a1fcdc08a01a3c2c54" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.886ex; height:2.843ex;" alt="{\displaystyle B(a,r)}"></span>, then Cauchy's estimate says:<sup id="cite_ref-1" class="reference"><a href="#cite_note-1">[1]</a></sup> for each integer <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n>0}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>n</mi>
<mo>></mo>
<mn>0</mn>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle n>0}</annotation>
</semantics>
</math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/27a6a5d982d54202a14f111cb8a49210501b2c96" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.656ex; height:2.176ex;" alt="{\displaystyle n>0}"></span>,
</p>
<dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle |f^{(n)}(a)|\leq {\frac {n!}{r^{n}}}M}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mrow class="MJX-TeXAtom-ORD">
<mo stretchy="false">|</mo>
</mrow>
<msup>
<mi>f</mi>
<mrow class="MJX-TeXAtom-ORD">
<mo stretchy="false">(</mo>
<mi>n</mi>
<mo stretchy="false">)</mo>
</mrow>
</msup>
<mo stretchy="false">(</mo>
<mi>a</mi>
<mo stretchy="false">)</mo>
<mrow class="MJX-TeXAtom-ORD">
<mo stretchy="false">|</mo>
</mrow>
<mo>≤<!-- ≤ --></mo>
<mrow class="MJX-TeXAtom-ORD">
<mfrac>
<mrow>
<mi>n</mi>
<mo>!</mo>
</mrow>
<msup>
<mi>r</mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>n</mi>
</mrow>
</msup>
</mfrac>
</mrow>
<mi>M</mi>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle |f^{(n)}(a)|\leq {\frac {n!}{r^{n}}}M}</annotation>
</semantics>
</math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/80ad645ad10f13304469925f6f04b70ef8141470" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:16.795ex; height:5.343ex;" alt="{\displaystyle |f^{(n)}(a)|\leq {\frac {n!}{r^{n}}}M}"></span></dd></dl>
<p>where <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f^{(n)}}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<msup>
<mi>f</mi>
<mrow class="MJX-TeXAtom-ORD">
<mo stretchy="false">(</mo>
<mi>n</mi>
<mo stretchy="false">)</mo>
</mrow>
</msup>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle f^{(n)}}</annotation>
</semantics>
</math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9dfb1963ccde0e87eb3838f51dc19041e2ff3816" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.818ex; height:3.176ex;" alt="{\displaystyle f^{(n)}}"></span> is the <i>n</i>-th <a href="/https/en.wikipedia.org/wiki/Complex_derivative" class="mw-redirect" title="Complex derivative">complex derivative</a> of <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>f</mi>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle f}</annotation>
</semantics>
</math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/132e57acb643253e7810ee9702d9581f159a1c61" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.279ex; height:2.509ex;" alt="{\displaystyle f}"></span>; i.e., <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f'={\frac {\partial f}{\partial z}}}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<msup>
<mi>f</mi>
<mo>′</mo>
</msup>
<mo>=</mo>
<mrow class="MJX-TeXAtom-ORD">
<mfrac>
<mrow>
<mi mathvariant="normal">∂<!-- ∂ --></mi>
<mi>f</mi>
</mrow>
<mrow>
<mi mathvariant="normal">∂<!-- ∂ --></mi>
<mi>z</mi>
</mrow>
</mfrac>
</mrow>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle f'={\frac {\partial f}{\partial z}}}</annotation>
</semantics>
</math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2a9000aecfae183657471bd560a8ee07ac2da6c8" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:8.536ex; height:5.676ex;" alt="{\displaystyle f'={\frac {\partial f}{\partial z}}}"></span> and <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f^{(n)}=(f^{(n-1)})^{'}}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<msup>
<mi>f</mi>
<mrow class="MJX-TeXAtom-ORD">
<mo stretchy="false">(</mo>
<mi>n</mi>
<mo stretchy="false">)</mo>
</mrow>
</msup>
<mo>=</mo>
<mo stretchy="false">(</mo>
<msup>
<mi>f</mi>
<mrow class="MJX-TeXAtom-ORD">
<mo stretchy="false">(</mo>
<mi>n</mi>
<mo>−<!-- − --></mo>
<mn>1</mn>
<mo stretchy="false">)</mo>
</mrow>
</msup>
<msup>
<mo stretchy="false">)</mo>
<mrow class="MJX-TeXAtom-ORD">
<msup>
<mi></mi>
<mo>′</mo>
</msup>
</mrow>
</msup>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle f^{(n)}=(f^{(n-1)})^{'}}</annotation>
</semantics>
</math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/31da025df7202b9cb0bec66ae999126d62d91963" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:15.409ex; height:3.343ex;" alt="{\displaystyle f^{(n)}=(f^{(n-1)})^{'}}"></span> (see <a href="/https/en.wikipedia.org/wiki/Wirtinger_derivatives#Relation_with_complex_differentiation" title="Wirtinger derivatives">Wirtinger derivatives § Relation with complex differentiation</a>).
</p><p>Moreover, taking <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f(z)=z^{n},a=0,r=1}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>f</mi>
<mo stretchy="false">(</mo>
<mi>z</mi>
<mo stretchy="false">)</mo>
<mo>=</mo>
<msup>
<mi>z</mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>n</mi>
</mrow>
</msup>
<mo>,</mo>
<mi>a</mi>
<mo>=</mo>
<mn>0</mn>
<mo>,</mo>
<mi>r</mi>
<mo>=</mo>
<mn>1</mn>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle f(z)=z^{n},a=0,r=1}</annotation>
</semantics>
</math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/67c7e70948200626ceb1399b4934004d4be3f9a6" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:22.451ex; height:2.843ex;" alt="{\displaystyle f(z)=z^{n},a=0,r=1}"></span> shows the above estimate cannot be improved.
</p><p>As a corollary, for example, we obtain <a href="/https/en.wikipedia.org/wiki/Liouville%27s_theorem_(complex_analysis)" title="Liouville's theorem (complex analysis)">Liouville's theorem</a>, which says a bounded entire function is constant (indeed, let <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle r\to \infty }">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>r</mi>
<mo stretchy="false">→<!-- → --></mo>
<mi mathvariant="normal">∞<!-- ∞ --></mi>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle r\to \infty }</annotation>
</semantics>
</math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/dcd3a85ea2e3d6b4027434e502cace4177d7a3e8" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:6.986ex; height:1.843ex;" alt="{\displaystyle r\to \infty }"></span> in the estimate.) Slightly more generally, if <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>f</mi>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle f}</annotation>
</semantics>
</math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/132e57acb643253e7810ee9702d9581f159a1c61" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.279ex; height:2.509ex;" alt="{\displaystyle f}"></span> is an entire function bounded by <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A+B|z|^{k}}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>A</mi>
<mo>+</mo>
<mi>B</mi>
<mrow class="MJX-TeXAtom-ORD">
<mo stretchy="false">|</mo>
</mrow>
<mi>z</mi>
<msup>
<mrow class="MJX-TeXAtom-ORD">
<mo stretchy="false">|</mo>
</mrow>
<mrow class="MJX-TeXAtom-ORD">
<mi>k</mi>
</mrow>
</msup>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle A+B|z|^{k}}</annotation>
</semantics>
</math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f465afd08072e87371a1b95edbb4b5e71839bce0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:9.818ex; height:3.343ex;" alt="{\displaystyle A+B|z|^{k}}"></span> for some constants <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A,B}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>A</mi>
<mo>,</mo>
<mi>B</mi>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle A,B}</annotation>
</semantics>
</math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/96c3298ea9aa77c226be56a7d8515baaa517b90b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:4.541ex; height:2.509ex;" alt="{\displaystyle A,B}"></span> and some integer <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle k>0}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>k</mi>
<mo>></mo>
<mn>0</mn>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle k>0}</annotation>
</semantics>
</math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/27b3af208b148139eefc03f0f80fa94c38c5af45" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.472ex; height:2.176ex;" alt="{\displaystyle k>0}"></span>, then <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>f</mi>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle f}</annotation>
</semantics>
</math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/132e57acb643253e7810ee9702d9581f159a1c61" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.279ex; height:2.509ex;" alt="{\displaystyle f}"></span> is a polynomial.<sup id="cite_ref-2" class="reference"><a href="#cite_note-2">[2]</a></sup>
</p>
<div class="mw-heading mw-heading2"><h2 id="Proof">Proof</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/https/en.wikipedia.org/w/index.php?title=Cauchy%27s_estimate&veaction=edit&section=2" title="Edit section: Proof" class="mw-editsection-visualeditor"><span>edit</span></a><span class="mw-editsection-divider"> | </span><a href="/https/en.wikipedia.org/w/index.php?title=Cauchy%27s_estimate&action=edit&section=2" title="Edit section's source code: Proof"><span>edit source</span></a><span class="mw-editsection-bracket">]</span></span></div>
<p>We start with <a href="/https/en.wikipedia.org/wiki/Cauchy%27s_integral_formula" title="Cauchy's integral formula">Cauchy's integral formula</a> applied to <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>f</mi>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle f}</annotation>
</semantics>
</math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/132e57acb643253e7810ee9702d9581f159a1c61" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.279ex; height:2.509ex;" alt="{\displaystyle f}"></span>
</p>
<dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f(z)={\frac {1}{2\pi i}}\int _{|w-a|=r'}{\frac {f(w)}{w-z}}\,dw,}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>f</mi>
<mo stretchy="false">(</mo>
<mi>z</mi>
<mo stretchy="false">)</mo>
<mo>=</mo>
<mrow class="MJX-TeXAtom-ORD">
<mfrac>
<mn>1</mn>
<mrow>
<mn>2</mn>
<mi>π<!-- π --></mi>
<mi>i</mi>
</mrow>
</mfrac>
</mrow>
<msub>
<mo>∫<!-- ∫ --></mo>
<mrow class="MJX-TeXAtom-ORD">
<mrow class="MJX-TeXAtom-ORD">
<mo stretchy="false">|</mo>
</mrow>
<mi>w</mi>
<mo>−<!-- − --></mo>
<mi>a</mi>
<mrow class="MJX-TeXAtom-ORD">
<mo stretchy="false">|</mo>
</mrow>
<mo>=</mo>
<msup>
<mi>r</mi>
<mo>′</mo>
</msup>
</mrow>
</msub>
<mrow class="MJX-TeXAtom-ORD">
<mfrac>
<mrow>
<mi>f</mi>
<mo stretchy="false">(</mo>
<mi>w</mi>
<mo stretchy="false">)</mo>
</mrow>
<mrow>
<mi>w</mi>
<mo>−<!-- − --></mo>
<mi>z</mi>
</mrow>
</mfrac>
</mrow>
<mspace width="thinmathspace" />
<mi>d</mi>
<mi>w</mi>
<mo>,</mo>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle f(z)={\frac {1}{2\pi i}}\int _{|w-a|=r'}{\frac {f(w)}{w-z}}\,dw,}</annotation>
</semantics>
</math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7bf5a924a9e38462b49516bbf43b93f6321ae2e2" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.671ex; width:30.841ex; height:6.509ex;" alt="{\displaystyle f(z)={\frac {1}{2\pi i}}\int _{|w-a|=r'}{\frac {f(w)}{w-z}}\,dw,}"></span></dd></dl>
<p>where <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle r'<r}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<msup>
<mi>r</mi>
<mo>′</mo>
</msup>
<mo><</mo>
<mi>r</mi>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle r'<r}</annotation>
</semantics>
</math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a0bfd384ece42a19de6855ed576bffb0d10a79d1" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.88ex; height:2.509ex;" alt="{\displaystyle r'<r}"></span>. By the <a href="/https/en.wikipedia.org/wiki/Differentiation_under_the_integral_sign" class="mw-redirect" title="Differentiation under the integral sign">differentiation under the integral sign</a> (in the complex variable),<sup id="cite_ref-3" class="reference"><a href="#cite_note-3">[3]</a></sup> we get:
</p>
<dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f^{(n)}(z)={\frac {n!}{2\pi i}}\int _{|w-a|=r'}{\frac {f(w)}{(w-z)^{n+1}}}\,dw.}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<msup>
<mi>f</mi>
<mrow class="MJX-TeXAtom-ORD">
<mo stretchy="false">(</mo>
<mi>n</mi>
<mo stretchy="false">)</mo>
</mrow>
</msup>
<mo stretchy="false">(</mo>
<mi>z</mi>
<mo stretchy="false">)</mo>
<mo>=</mo>
<mrow class="MJX-TeXAtom-ORD">
<mfrac>
<mrow>
<mi>n</mi>
<mo>!</mo>
</mrow>
<mrow>
<mn>2</mn>
<mi>π<!-- π --></mi>
<mi>i</mi>
</mrow>
</mfrac>
</mrow>
<msub>
<mo>∫<!-- ∫ --></mo>
<mrow class="MJX-TeXAtom-ORD">
<mrow class="MJX-TeXAtom-ORD">
<mo stretchy="false">|</mo>
</mrow>
<mi>w</mi>
<mo>−<!-- − --></mo>
<mi>a</mi>
<mrow class="MJX-TeXAtom-ORD">
<mo stretchy="false">|</mo>
</mrow>
<mo>=</mo>
<msup>
<mi>r</mi>
<mo>′</mo>
</msup>
</mrow>
</msub>
<mrow class="MJX-TeXAtom-ORD">
<mfrac>
<mrow>
<mi>f</mi>
<mo stretchy="false">(</mo>
<mi>w</mi>
<mo stretchy="false">)</mo>
</mrow>
<mrow>
<mo stretchy="false">(</mo>
<mi>w</mi>
<mo>−<!-- − --></mo>
<mi>z</mi>
<msup>
<mo stretchy="false">)</mo>
<mrow class="MJX-TeXAtom-ORD">
<mi>n</mi>
<mo>+</mo>
<mn>1</mn>
</mrow>
</msup>
</mrow>
</mfrac>
</mrow>
<mspace width="thinmathspace" />
<mi>d</mi>
<mi>w</mi>
<mo>.</mo>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle f^{(n)}(z)={\frac {n!}{2\pi i}}\int _{|w-a|=r'}{\frac {f(w)}{(w-z)^{n+1}}}\,dw.}</annotation>
</semantics>
</math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b02f9212a37b2971062484c1bd45e0ede21feba6" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.671ex; width:38.509ex; height:6.509ex;" alt="{\displaystyle f^{(n)}(z)={\frac {n!}{2\pi i}}\int _{|w-a|=r'}{\frac {f(w)}{(w-z)^{n+1}}}\,dw.}"></span></dd></dl>
<p>Thus,
</p>
<dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle |f^{(n)}(a)|\leq {\frac {n!M}{2\pi }}\int _{|w-a|=r'}{\frac {|dw|}{|w-a|^{n+1}}}={\frac {n!M}{{r'}^{n}}}.}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mrow class="MJX-TeXAtom-ORD">
<mo stretchy="false">|</mo>
</mrow>
<msup>
<mi>f</mi>
<mrow class="MJX-TeXAtom-ORD">
<mo stretchy="false">(</mo>
<mi>n</mi>
<mo stretchy="false">)</mo>
</mrow>
</msup>
<mo stretchy="false">(</mo>
<mi>a</mi>
<mo stretchy="false">)</mo>
<mrow class="MJX-TeXAtom-ORD">
<mo stretchy="false">|</mo>
</mrow>
<mo>≤<!-- ≤ --></mo>
<mrow class="MJX-TeXAtom-ORD">
<mfrac>
<mrow>
<mi>n</mi>
<mo>!</mo>
<mi>M</mi>
</mrow>
<mrow>
<mn>2</mn>
<mi>π<!-- π --></mi>
</mrow>
</mfrac>
</mrow>
<msub>
<mo>∫<!-- ∫ --></mo>
<mrow class="MJX-TeXAtom-ORD">
<mrow class="MJX-TeXAtom-ORD">
<mo stretchy="false">|</mo>
</mrow>
<mi>w</mi>
<mo>−<!-- − --></mo>
<mi>a</mi>
<mrow class="MJX-TeXAtom-ORD">
<mo stretchy="false">|</mo>
</mrow>
<mo>=</mo>
<msup>
<mi>r</mi>
<mo>′</mo>
</msup>
</mrow>
</msub>
<mrow class="MJX-TeXAtom-ORD">
<mfrac>
<mrow>
<mrow class="MJX-TeXAtom-ORD">
<mo stretchy="false">|</mo>
</mrow>
<mi>d</mi>
<mi>w</mi>
<mrow class="MJX-TeXAtom-ORD">
<mo stretchy="false">|</mo>
</mrow>
</mrow>
<mrow>
<mrow class="MJX-TeXAtom-ORD">
<mo stretchy="false">|</mo>
</mrow>
<mi>w</mi>
<mo>−<!-- − --></mo>
<mi>a</mi>
<msup>
<mrow class="MJX-TeXAtom-ORD">
<mo stretchy="false">|</mo>
</mrow>
<mrow class="MJX-TeXAtom-ORD">
<mi>n</mi>
<mo>+</mo>
<mn>1</mn>
</mrow>
</msup>
</mrow>
</mfrac>
</mrow>
<mo>=</mo>
<mrow class="MJX-TeXAtom-ORD">
<mfrac>
<mrow>
<mi>n</mi>
<mo>!</mo>
<mi>M</mi>
</mrow>
<msup>
<mrow class="MJX-TeXAtom-ORD">
<msup>
<mi>r</mi>
<mo>′</mo>
</msup>
</mrow>
<mrow class="MJX-TeXAtom-ORD">
<mi>n</mi>
</mrow>
</msup>
</mfrac>
</mrow>
<mo>.</mo>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle |f^{(n)}(a)|\leq {\frac {n!M}{2\pi }}\int _{|w-a|=r'}{\frac {|dw|}{|w-a|^{n+1}}}={\frac {n!M}{{r'}^{n}}}.}</annotation>
</semantics>
</math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/801b61869a791e0216cbda46bec9ff278218af5c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:45.908ex; height:6.843ex;" alt="{\displaystyle |f^{(n)}(a)|\leq {\frac {n!M}{2\pi }}\int _{|w-a|=r'}{\frac {|dw|}{|w-a|^{n+1}}}={\frac {n!M}{{r'}^{n}}}.}"></span></dd></dl>
<p>Letting <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle r'\to r}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<msup>
<mi>r</mi>
<mo>′</mo>
</msup>
<mo stretchy="false">→<!-- → --></mo>
<mi>r</mi>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle r'\to r}</annotation>
</semantics>
</math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8e8bf8c1abd5104a9f3147a1ff09513f4c4e99a6" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:6.396ex; height:2.509ex;" alt="{\displaystyle r'\to r}"></span> finishes the proof. <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \square }">
<semantics>
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<mstyle displaystyle="true" scriptlevel="0">
<mi>◻<!-- ◻ --></mi>
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<annotation encoding="application/x-tex">{\displaystyle \square }</annotation>
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</math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/455831d58fa08f311b934d324adcff89a868b4e4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.808ex; height:2.176ex;" alt="{\displaystyle \square }"></span>
</p>
<div class="mw-heading mw-heading2"><h2 id="Related_estimate">Related estimate</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/https/en.wikipedia.org/w/index.php?title=Cauchy%27s_estimate&veaction=edit&section=3" title="Edit section: Related estimate" class="mw-editsection-visualeditor"><span>edit</span></a><span class="mw-editsection-divider"> | </span><a href="/https/en.wikipedia.org/w/index.php?title=Cauchy%27s_estimate&action=edit&section=3" title="Edit section's source code: Related estimate"><span>edit source</span></a><span class="mw-editsection-bracket">]</span></span></div>
<p>Here is a somehow more general but slightly less precise estimate. It says:<sup id="cite_ref-4" class="reference"><a href="#cite_note-4">[4]</a></sup> given an open subset <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle U\subset \mathbb {C} }">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>U</mi>
<mo>⊂<!-- ⊂ --></mo>
<mrow class="MJX-TeXAtom-ORD">
<mi mathvariant="double-struck">C</mi>
</mrow>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle U\subset \mathbb {C} }</annotation>
</semantics>
</math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1987e3c9fabe04333aa33cf0452f94a6daa71cbd" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:6.559ex; height:2.176ex;" alt="{\displaystyle U\subset \mathbb {C} }"></span>, a compact subset <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle K\subset U}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>K</mi>
<mo>⊂<!-- ⊂ --></mo>
<mi>U</mi>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle K\subset U}</annotation>
</semantics>
</math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/535ed682dca3a92174f9efae55ca7ddd073567c6" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:6.947ex; height:2.176ex;" alt="{\displaystyle K\subset U}"></span> and an integer <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n>0}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>n</mi>
<mo>></mo>
<mn>0</mn>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle n>0}</annotation>
</semantics>
</math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/27a6a5d982d54202a14f111cb8a49210501b2c96" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.656ex; height:2.176ex;" alt="{\displaystyle n>0}"></span>, there is a constant <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle C_{K,n}}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<msub>
<mi>C</mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>K</mi>
<mo>,</mo>
<mi>n</mi>
</mrow>
</msub>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle C_{K,n}}</annotation>
</semantics>
</math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/af912029b504008bf944ea5cc879eb729560c539" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:4.799ex; height:2.843ex;" alt="{\displaystyle C_{K,n}}"></span> such that
</p>
<dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \sup _{K}|f^{(n)}|\leq C_{K,n}\int _{U}|f|\,d\mu }">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<munder>
<mo movablelimits="true" form="prefix">sup</mo>
<mrow class="MJX-TeXAtom-ORD">
<mi>K</mi>
</mrow>
</munder>
<mrow class="MJX-TeXAtom-ORD">
<mo stretchy="false">|</mo>
</mrow>
<msup>
<mi>f</mi>
<mrow class="MJX-TeXAtom-ORD">
<mo stretchy="false">(</mo>
<mi>n</mi>
<mo stretchy="false">)</mo>
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</msup>
<mrow class="MJX-TeXAtom-ORD">
<mo stretchy="false">|</mo>
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<mo>≤<!-- ≤ --></mo>
<msub>
<mi>C</mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>K</mi>
<mo>,</mo>
<mi>n</mi>
</mrow>
</msub>
<msub>
<mo>∫<!-- ∫ --></mo>
<mrow class="MJX-TeXAtom-ORD">
<mi>U</mi>
</mrow>
</msub>
<mrow class="MJX-TeXAtom-ORD">
<mo stretchy="false">|</mo>
</mrow>
<mi>f</mi>
<mrow class="MJX-TeXAtom-ORD">
<mo stretchy="false">|</mo>
</mrow>
<mspace width="thinmathspace" />
<mi>d</mi>
<mi>μ<!-- μ --></mi>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle \sup _{K}|f^{(n)}|\leq C_{K,n}\int _{U}|f|\,d\mu }</annotation>
</semantics>
</math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4b1993403dc123658eeb2df7db7f2d2b654f5510" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:26.034ex; height:5.843ex;" alt="{\displaystyle \sup _{K}|f^{(n)}|\leq C_{K,n}\int _{U}|f|\,d\mu }"></span></dd></dl>
<p>where <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle d\mu }">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>d</mi>
<mi>μ<!-- μ --></mi>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle d\mu }</annotation>
</semantics>
</math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/afc73ac32e066a58718ee0c9576b05f7485ab008" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:2.618ex; height:2.676ex;" alt="{\displaystyle d\mu }"></span> is the Lebesgue measure.
</p>
<div class="mw-heading mw-heading2"><h2 id="See_also">See also</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/https/en.wikipedia.org/w/index.php?title=Cauchy%27s_estimate&veaction=edit&section=4" title="Edit section: See also" class="mw-editsection-visualeditor"><span>edit</span></a><span class="mw-editsection-divider"> | </span><a href="/https/en.wikipedia.org/w/index.php?title=Cauchy%27s_estimate&action=edit&section=4" title="Edit section's source code: See also"><span>edit source</span></a><span class="mw-editsection-bracket">]</span></span></div>
<ul><li><a href="/https/en.wikipedia.org/wiki/Taylor%27s_theorem" title="Taylor's theorem">Taylor's theorem</a></li></ul>
<div class="mw-heading mw-heading2"><h2 id="References">References</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/https/en.wikipedia.org/w/index.php?title=Cauchy%27s_estimate&veaction=edit&section=5" title="Edit section: References" class="mw-editsection-visualeditor"><span>edit</span></a><span class="mw-editsection-divider"> | </span><a href="/https/en.wikipedia.org/w/index.php?title=Cauchy%27s_estimate&action=edit&section=5" title="Edit section's source code: References"><span>edit source</span></a><span class="mw-editsection-bracket">]</span></span></div>
<style data-mw-deduplicate="TemplateStyles:r1217336898">.mw-parser-output .reflist{font-size:90%;margin-bottom:0.5em;list-style-type:decimal}.mw-parser-output .reflist .references{font-size:100%;margin-bottom:0;list-style-type:inherit}.mw-parser-output .reflist-columns-2{column-width:30em}.mw-parser-output .reflist-columns-3{column-width:25em}.mw-parser-output .reflist-columns{margin-top:0.3em}.mw-parser-output .reflist-columns ol{margin-top:0}.mw-parser-output .reflist-columns li{page-break-inside:avoid;break-inside:avoid-column}.mw-parser-output .reflist-upper-alpha{list-style-type:upper-alpha}.mw-parser-output .reflist-upper-roman{list-style-type:upper-roman}.mw-parser-output .reflist-lower-alpha{list-style-type:lower-alpha}.mw-parser-output .reflist-lower-greek{list-style-type:lower-greek}.mw-parser-output .reflist-lower-roman{list-style-type:lower-roman}</style><div class="reflist">
<div class="mw-references-wrap"><ol class="references">
<li id="cite_note-1"><span class="mw-cite-backlink"><b><a href="#cite_ref-1">^</a></b></span> <span class="reference-text"><a href="#CITEREFRudin1986">Rudin 1986</a>, Theorem 10.26.</span>
</li>
<li id="cite_note-2"><span class="mw-cite-backlink"><b><a href="#cite_ref-2">^</a></b></span> <span class="reference-text"><a href="#CITEREFRudin1986">Rudin 1986</a>, Ch 10. Exercise 4.</span>
</li>
<li id="cite_note-3"><span class="mw-cite-backlink"><b><a href="#cite_ref-3">^</a></b></span> <span class="reference-text">This step is Exercise 7 in Ch. 10. of <a href="#CITEREFRudin1986">Rudin 1986</a></span>
</li>
<li id="cite_note-4"><span class="mw-cite-backlink"><b><a href="#cite_ref-4">^</a></b></span> <span class="reference-text"><a href="#CITEREFHörmander1990">Hörmander 1990</a><span class="error harv-error" style="display: none; font-size:100%"> harvnb error: no target: CITEREFHörmander1990 (<a href="/https/en.wikipedia.org/wiki/Category:Harv_and_Sfn_template_errors" title="Category:Harv and Sfn template errors">help</a>)</span></span>
</li>
</ol></div></div>
<ul><li><style data-mw-deduplicate="TemplateStyles:r1215172403">.mw-parser-output cite.citation{font-style:inherit;word-wrap:break-word}.mw-parser-output .citation q{quotes:"\"""\"""'""'"}.mw-parser-output .citation:target{background-color:rgba(0,127,255,0.133)}.mw-parser-output .id-lock-free.id-lock-free a{background:url("//upload.wikimedia.org/wikipedia/commons/6/65/Lock-green.svg")right 0.1em center/9px no-repeat}body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-free a{background-size:contain}.mw-parser-output .id-lock-limited.id-lock-limited a,.mw-parser-output .id-lock-registration.id-lock-registration a{background:url("//upload.wikimedia.org/wikipedia/commons/d/d6/Lock-gray-alt-2.svg")right 0.1em center/9px no-repeat}body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-limited a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-registration a{background-size:contain}.mw-parser-output .id-lock-subscription.id-lock-subscription a{background:url("//upload.wikimedia.org/wikipedia/commons/a/aa/Lock-red-alt-2.svg")right 0.1em center/9px no-repeat}body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-subscription a{background-size:contain}.mw-parser-output .cs1-ws-icon a{background:url("//upload.wikimedia.org/wikipedia/commons/4/4c/Wikisource-logo.svg")right 0.1em center/12px no-repeat}body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .cs1-ws-icon a{background-size:contain}.mw-parser-output .cs1-code{color:inherit;background:inherit;border:none;padding:inherit}.mw-parser-output .cs1-hidden-error{display:none;color:#d33}.mw-parser-output .cs1-visible-error{color:#d33}.mw-parser-output .cs1-maint{display:none;color:#2C882D;margin-left:0.3em}.mw-parser-output .cs1-format{font-size:95%}.mw-parser-output .cs1-kern-left{padding-left:0.2em}.mw-parser-output .cs1-kern-right{padding-right:0.2em}.mw-parser-output .citation .mw-selflink{font-weight:inherit}html.skin-theme-clientpref-night .mw-parser-output .cs1-maint{color:#18911F}html.skin-theme-clientpref-night .mw-parser-output .cs1-visible-error,html.skin-theme-clientpref-night .mw-parser-output .cs1-hidden-error{color:#f8a397}@media(prefers-color-scheme:dark){html.skin-theme-clientpref-os .mw-parser-output .cs1-visible-error,html.skin-theme-clientpref-os .mw-parser-output .cs1-hidden-error{color:#f8a397}html.skin-theme-clientpref-os .mw-parser-output .cs1-maint{color:#18911F}}</style><cite id="CITEREFLars_Hörmander1990" class="citation cs2"><a href="/https/en.wikipedia.org/wiki/Lars_H%C3%B6rmander" title="Lars Hörmander">Lars Hörmander</a> (1990) [1966], <a rel="nofollow" class="external text" href="https://books.google.com/books?id=MaM7AAAAQBAJ"><i>An Introduction to Complex Analysis in Several Variables</i></a> (3rd ed.), North Holland, <a href="/https/en.wikipedia.org/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/https/en.wikipedia.org/wiki/Special:BookSources/978-1-493-30273-4" title="Special:BookSources/978-1-493-30273-4"><bdi>978-1-493-30273-4</bdi></a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=An+Introduction+to+Complex+Analysis+in+Several+Variables&rft.edition=3rd&rft.pub=North+Holland&rft.date=1990&rft.isbn=978-1-493-30273-4&rft.au=Lars+H%C3%B6rmander&rft_id=https%3A%2F%2Ffanyv88.com%3A443%2Fhttps%2Fbooks.google.com%2Fbooks%3Fid%3DMaM7AAAAQBAJ&rfr_id=info%3Asid%2Fen.wikipedia.org%3ACauchy%27s+estimate" class="Z3988"></span></li>
<li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1215172403"><cite id="CITEREFRudin1986" class="citation book cs1"><a href="/https/en.wikipedia.org/wiki/Walter_Rudin" title="Walter Rudin">Rudin, Walter</a> (1986). <i>Real and Complex Analysis (International Series in Pure and Applied Mathematics)</i>. McGraw-Hill. <a href="/https/en.wikipedia.org/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/https/en.wikipedia.org/wiki/Special:BookSources/978-0-07-054234-1" title="Special:BookSources/978-0-07-054234-1"><bdi>978-0-07-054234-1</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Real+and+Complex+Analysis+%28International+Series+in+Pure+and+Applied+Mathematics%29&rft.pub=McGraw-Hill&rft.date=1986&rft.isbn=978-0-07-054234-1&rft.aulast=Rudin&rft.aufirst=Walter&rfr_id=info%3Asid%2Fen.wikipedia.org%3ACauchy%27s+estimate" class="Z3988"></span></li></ul>
<div class="mw-heading mw-heading2"><h2 id="Further_reading">Further reading</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/https/en.wikipedia.org/w/index.php?title=Cauchy%27s_estimate&veaction=edit&section=6" title="Edit section: Further reading" class="mw-editsection-visualeditor"><span>edit</span></a><span class="mw-editsection-divider"> | </span><a href="/https/en.wikipedia.org/w/index.php?title=Cauchy%27s_estimate&action=edit&section=6" title="Edit section's source code: Further reading"><span>edit source</span></a><span class="mw-editsection-bracket">]</span></span></div>
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Whether or not the change was made through a Tor exit node (tor_exit_node) | false |
Unix timestamp of change (timestamp) | '1722241281' |